Preheat oven to 375 °F (190 °C). Grease a 9×9-inch (23×23 cm) baking dish with butter.

In a medium saucepan over medium heat, melt butter. Add garlic and sauté 1 minute until fragrant.

Stir in flour and cook for another minute, whisking constantly to form a roux.

Gradually whisk in milk and cream until smooth. Bring to a gentle simmer, then whisk in Dijon mustard, salt, and pepper.

Remove sauce from heat and stir in 1 cup of the cheddar and half of the Gruyère until the cheeses melt into the sauce.

Arrange half of the potato slices in an even layer in the prepared baking dish. Pour half of the cheese sauce over the potatoes.

Add the remaining potato slices, then pour the rest of the sauce on top.

Sprinkle the remaining cheddar and Gruyère evenly over the top layer.

Cover with foil and bake for 30 minutes. Remove foil and bake 15–20 more minutes, until golden and bubbling and potatoes are tender when pierced with a fork.

Let rest 5 minutes before serving, then garnish with fresh chives if desired.